Efron's Dice are a set of four (non-standard) dice A, B, C, D such that A has a 2/3 probability of beating B when they are rolled together; B of C; C of D; and D has a 2/3 probability of beating A. The dice are thus non-transitive.

A: 4/4/4/4/0/0

B: 3/3/3/3/3/3

C: 6/6/2/2/2/2

D: 5/5/5/1/1/1

The probability of A beating B, and B of C; is clearly 2/3. For C over D the probability is

1/3+2/3×1/2 = 2/3;

for D over A it is

1/2 + 1/2×1/3 = 2/3.

External links

• MathWorld page